What the article is discussing is how methylation differs between very young and very old people. The abstract of the original paper may be more instructive:

Human aging cannot be fully understood in terms of the constrained genetic setting. Epigenetic drift is an alternative means of explaining age-associated alterations. To address this issue, we performed whole-genome bisulfite sequencing (WGBS) of newborn and centenarian genomes. The centenarian DNA had a lower DNA methylation content and a reduced correlation in the methylation status of neighboring cytosine—phosphate—guanine (CpGs) throughout the genome in comparison with the more homogeneously methylated newborn DNA. The more hypomethylated CpGs observed in the centenarian DNA compared with the neonate covered all genomic compartments, such as promoters, exonic, intronic, and intergenic regions. For regulatory regions, the most hypomethylated sequences in the centenarian DNA were present mainly at CpG-poor promoters and in tissue-specific genes, whereas a greater level of DNA methylation was observed in CpG island promoters. We extended the study to a larger cohort of newborn and nonagenarian samples using a 450,000 CpG-site DNA methylation microarray that reinforced the observation of more hypomethylated DNA sequences in the advanced age group. WGBS and 450,000 analyses of middle-age individuals demonstrated DNA methylomes in the crossroad between the newborn and the nonagenarian/centenarian groups. Our study constitutes a unique DNA methylation analysis of the extreme points of human life at a single-nucleotide resolution level.

what I understand from that wall of text is this: The paper puts forward is another factor that contributes to errors cropping up causing diseases associated with old age, like cancer. Methylation controls (or should that be retards?) transcriptional activity, so a change in methylation patterns, or a drop in the occurence of methylation, would change the types of activities the DNA undergoes, and change the probability (probably upwards) of things going wrong.

I am but a lowly undergrad who doesn't pay as much attention is lectures as he should, so please someone correct me if I am wrong.

It means that although the DNA will remain the same, how it is transcripted into proteins (and how often, and in reaction to what stimuli) changes. That is the purview of epigenetic study (the epi- prefix meaning over or above). Here's a useful link [wikipedia.org].

As we age, the core of our biological being — the sequence of our DNA, which makes up our genes — remains the same.

This was falsified several years ago when it was shown that retrotranspons alter the sequence of DNA in each cell dynamically continuously. Not only that, but cells are altered differently, so a person's cells diverge as they age. The paper is usually paywalled but I have a copy thanks to the generosity of the authors, if anyone wants a copy.

Sorry, but as a matter of principle I automatically reject any claim that has as its central tenant a theory that has already been falsified. Keep up or keep the hell out.